Average Velocity and Average Speed

The velocity-time graph of a particle in one-dimensional motion is shown in Figure.

Which of the following formulae are correct for describing the motion of the particle over the time-interval ${\mathrm{t}}_2$ to ${\mathrm{t}}_1$?

Figure given shows the $x-t$ plot of a particle in one-dimensional motion. Three different equal intervals of time are shown. In which interval is the average speed greatest, and in which is it the least? Give the sign of average velocity for each interval.

A man walks on a straight road from his home to a market $2.5 \mathrm{km}$ away with a speed of $5 \mathrm{k}\mathrm{m} {\mathrm{h}}^{-1}$. Finding the market closed, he instantly turns and walks back home with a speed of $7.5 \mathrm{k}\mathrm{m} {\mathrm{h}}^{-1}$. What is the
(a) magnitude of average velocity, and
(b) average speed of the man over the interval of time (i) $0$ to $30$ min, (ii) $0$ to $50$ min, (iii) $0$ to $40$ min?
[Note: You will appreciate from this exercise why it is better to define average speed as total path length divided by time, and not as the magnitude of average velocity. You would not like to tell the tired man on his return home that his average speed was zero!]

Explain clearly, with examples, the distinction between:
(a) magnitude of displacement (sometimes called distance) over an interval of time, and the total length of path covered by a particle over the same interval;
(b) magnitude of average velocity over an interval of time, and the average speed over the same interval. [Average speed of a particle over an interval of time is defined as the total path length divided by the time interval]. Show in both (a) and (b) that the second quantity is either greater than or equal to the first.
When is the equality sign true? [For simplicity, consider one-dimensional motion only].

A woman starts from her home at $9.00 \mathrm{am}$, walks with a speed of $5 \mathrm{km} {\mathrm{h}}^{-1}$ on a straight road up to her office $2.5 \mathrm{km}$ away, stays at office till $5.00 \mathrm{pm}$, and returns home by an auto with a speed of $25 \mathrm{km} {\mathrm{h}}^{-1}$. Choose suitable scale and plot the $\mathrm{x}-\mathrm{t}$ graph of her motion.